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vec(u)=(8,-3) Find the direction angle of vec(u). Enter your answer as an angle in degrees between 0^(@) and 360^(@) rounded to the nearest hundredth. theta=◻" 。 "

u⃗=(8,−3) \vec{u}=(8,-3) \newlineFind the direction angle of u⃗ \vec{u} . Enter your answer as an angle in degrees between 0∘ 0^{\circ} and 360∘ 360^{\circ} rounded to the nearest hundredth.\newlineθ=□∘ \theta= \square^{\circ}

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Inverses of sin, cos, and tan: degreesright chevron icon

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Q. u⃗=(8,−3) \vec{u}=(8,-3) \newlineFind the direction angle of u⃗ \vec{u} . Enter your answer as an angle in degrees between 0∘ 0^{\circ} and 360∘ 360^{\circ} rounded to the nearest hundredth.\newlineθ=□∘ \theta= \square^{\circ}
  1. Finding the Direction Angle: To find the direction angle of the vector u⃗=(8,−3)\vec{u} = (8, -3), we need to calculate the angle θ\theta that the vector makes with the positive x-axis. The direction angle can be found using the arctangent function (tan⁡−1\tan^{-1}), which gives us the angle in radians. We can then convert this angle to degrees. The formula to find the angle is θ=tan⁡−1(yx)\theta = \tan^{-1}(\frac{y}{x}), where xx and yy are the components of the vector.
  2. Plugging in the Components: First, we plug the components of u⃗\vec{u} into the formula: θ=tan⁡−1(−38)\theta = \tan^{-1}\left(\frac{-3}{8}\right). This will give us the angle in radians.
  3. Calculating the Angle in Radians: Using a calculator, we find that θ=tan⁡−1(−38)≈−20.556\theta = \tan^{-1}(-\frac{3}{8}) \approx -20.556 degrees. However, this angle is not between 0∘0^\circ and 360∘360^\circ, so we need to adjust it.
  4. Converting the Angle to Degrees: Since the vector is in the fourth quadrant (because xx is positive and yy is negative), we add 360∘360^\circ to the angle to find the direction angle in the specified range. Therefore, the direction angle is −20.556∘+360∘=339.444∘-20.556^\circ + 360^\circ = 339.444^\circ.
  5. Adjusting the Angle: We round the direction angle to the nearest hundredth, which gives us θ≈339.44∘\theta \approx 339.44^\circ.

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